Spectral properties of hierarchical
نویسنده
چکیده
In this text we are interested in spectral properties of discrete Laplace operators deened on lattices based on nitely-ramiied self-similar sets. The basic example is the lattice based on the Sierpinski gasket. We introduce a new renormalization map which appears to be a rational self-map of a compact complex manifolds. We relate some characteristics of its dynamics with some characteristics of the spectrum of our operator. More speciically, we give an explicite formula for the density of states in terms of the Green current of the map, and we relate the indeterminacy points of the map with the so-called Neuman-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map we can prove drastic diierent spectral properties of the operator. Hence, this work aims at a generalization and a better understanding of the initial work of physisits Rammal and Toulouse on the Sierpinski gasket (cf 31], 30]).
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